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GBM, HGBM

Syntax

GBM(L:LIST):IDEAL
HGBM(L:LIST):IDEAL
    

Summary

intersection of ideals for zero-dimensional schemes

Description

These functions compute the intersection of ideals corresponding to zero-dimensional schemes: GBM is for affine schemes, and HGBM for projective schemes. The list L must be a list of ideals. The function IntersectionList should be used for computing the intersection of a collection of general ideals.

The name GBM comes from the name of the algorithm used: Generalized Buchberger-Moeller. The prefix H comes from Homogeneous since ideals of projective schemes are necessarily homogeneous.

example

    
Use Q[x,y,z];
I1:=IdealOfPoints([[1,2,1], [0,1,0]]);     -- a simple affine scheme
I2:=IdealOfPoints([[1,1,1], [2,0,1]])^2;   -- another affine scheme
GBM([I1,I2]);                              -- intersect the ideals
Ideal(xz + yz - z^2 - x - y + 1,
 z^3 - 2z^2 + z,
 yz^2 - 2yz - z^2 + y + 2z - 1,
 y^2z - y^2 - yz + y,
 xy^2 + y^3 - 2x^2 - 5xy - 5y^2 + 2z^2 + 8x + 10y - 4z - 6,
 x^2y - y^3 + 2x^2 + 2xy + 4y^2 - 3z^2 - 8x - 8y + 6z + 5,
 x^3 + y^3 - 7x^2 - 5xy - 4y^2 + 5z^2 + 16x + 10y - 10z - 7,
 y^4 - 2y^3 - 4x^2 - 8xy - 3y^2 + 4z^2 + 16x + 16y - 8z - 12)
-------------------------------

Use Q[x[0..2]];
I1:=IdealOfProjectivePoints([[1,2,1], [0,1,0]]);   -- simple projective scheme
I2:=IdealOfProjectivePoints([[1,1,1], [2,0,1]])^2; -- another projective scheme
HGBM([I1,I2]);                                     -- intersect the ideals
Ideal(x[0]^3 - x[0]x[1]^2 - 5x[0]^2x[2] + x[1]^2x[2] + 8x[0]x[2]^2 - 4x[2]^3,
 x[0]^2x[1] + x[0]x[1]^2 - 3x[0]x[1]x[2] - x[1]^2x[2] + 2x[1]x[2]^2,
 x[0]x[1]^3 - 2x[0]^2x[2]^2 - 5x[0]x[1]x[2]^2 - 4x[1]^2x[2]^2 +
8x[0]x[2]^3 + 10x[1]x[2]^3 - 8x[2]^4,
 x[0]x[1]^2x[2] + x[1]^3x[2] - 2x[0]^2x[2]^2 - 5x[0]x[1]x[2]^2 
- 5x[1]^2x[2]^2 + 8x[0]x[2]^3 + 10x[1]x[2]^3 - 8x[2]^4,
 x[1]^4x[2] - 2x[1]^3x[2]^2 - 4x[0]^2x[2]^3 - 8x[0]x[1]x[2]^3 
- 3x[1]^2x[2]^3 + 16x[0]x[2]^4 + 16x[1]x[2]^4 - 16x[2]^5)
-------------------------------
        
    

See Also